The hostage-taker at a kosher supermarket in eastern Paris has been killed, according to Le Monde.
The newspaper quoted a police official saying that the man was believed to have links to the same Islamist group as the two suspects in the Charlie Hebdo massacre. They have also been killed.
French television images showed some people running out of the supermarket in eastern Paris. However, a police source said at least four other hostages had been killed.
An Israeli source said 15 hostages were rescued.
The official, recounting a conversation between Prime Minister Benjamin Netanyahu and the French president, said four hostages were killed and 15 freed.
French president Francois Hollande, speaking shortly afterwards in a national address, confirmed the four deaths.
Police forces stormed the supermarket this afternoon after the gunman threatened to kill the hostages if police launch an assault on the cornered brothers suspected in the newspaper massacre earlier this week, a police official said.
The official described the dual hostage situations as "clearly linked".
Several people had been wounded when the gunman opened fire in the market this afternoon and were able to flee and get medical care, the official said.
It was not clear whether there were other wounded inside the market, or whether the woman listed as the gunman's accomplice in a police bulletin was inside with him.
The exact number of hostages taken was unclear. Local media spoke of at least five. A police source said the man was equipped with automatic weapons.
Police immediately cordoned off the area and a helicopter was flying overhead.
North of the capital French anti-terrorist police sealed off a town and helicopters hovered overhead after a police source said the two men believed to have carried Wednesday's attack on the satirical Charlie Hebdo newspaper took at least one person hostage in a small print works.
Separately, Paris police named a man they were looking for in connection with Thursday's killing of a policewoman as 32-year-old Amedy Coulibaly. They said were also looking for a 26 year-old woman called Hayat Boumeddiene. They described both as armed and dangerous.